Sunday, May 29, 2011

A skateboarder hangs on to a merry-go-round turning at a constant rate. At a given instant, he lets go, and travels in a straight line at a constant speed, until he runs into a (straight) wall a given distance away. How much time does it take him to hit the fence?

As I read Victor's essay, he has three main critiques of the problem:

The problem is contrived, and fundamentally uninteresting (who would care how long it takes to hit the wall?), especially with the original data in the problem (all triangles are 30-60-90, etc., etc.).

The solution relies entirely on symbolic mathematics, and so is hard to generalize to other situations, e.g. varying speeds, curved walls, etc.

A more interesting--and, for many students, more valuable--approach would be to simulate the situation on a computer, so that students can appreciate how changing the parameters in the problem affects the answer.

While the first of these is valid, I think the second and third miss the point entirely, for reasons I'll explain.

Criticism #1: The problem is contrived and boring. I think Victor's right about this, and an analysis that stops at "it takes 4.7 seconds to hit the wall" isn't necessarily worth the time it takes to get there. But used as a springboard for generalization and extension -- I'm imagining one of Dan Meyer's WCYDWT videos -- this could be really cool. In fact, Victor's own suggested rephrasing of the problem, "What release point minimizes the time to hit the wall?" is one obvious extension.

Here's another approach that we don't ordinarily take: start with the general problem in roughly the way that I've stated it above, and have students make it specific enough to solve as a first approximation to a general solution. The task is now "Can we (always) compute the time it takes for the skateboarder to hit the wall?" and the students--much like mathematicians or engineers in real life--start by filling in the variables with plausible initial conditions to see what a solution might look like, before attempting a solution (or a model) with variables.

Criticisms #2 and #3 seem weird to me, for three reasons:

One goal of mathematics is the creation (or identification) of general, true statements. Victor's proposed simulation doesn't really allow for those. At a philosophical level, it could be argued that the simulator itself is equivalent to a set of true statements that describe all possible behaviors of the system, but without some way of tying those together, we lose what philosophers would call the "propositionality" of mathematics. Victor is claiming that our goal should not be a single answer or set of answers, but rather a computational model that allows us to find answers given a particular set of initial conditions. I'll concede that in some areas of mathematics--dynamical systems, for example--often such models are the best we can hope for. But even in those we want to make general statements about a system's behavior, and it's hard to imagine how we might do that without invoking symblic mathematics.

If our goal is to empower students to do more and more interesting mathematics, we can't just hand them simulators and tell them to go play: we need to teach them how to create those simulators. Doing that requires a lot of math and a lot of programming. So Victor's "simulation" model of doing math ultimately requires teaching kids a lot of traditional mathematics.

Even if we do give students a simulation that they can use to find a particular outcome, I think it's natural for students to ask "Why?" and "How can we be sure?" Those questions ask for justification and generalization that require ... traditional symbolic mathematics.

I think simulations are terrific ways for students to experiment, to learn, and to generate and answer interesting questions. But what we need to do is not replace traditional math with simulations, but use the power of models and simulations as an incentive to do, test, and generate exciting mathematics.

I have an activity about polyominos. I begin by reminding my students what a domino is. Then I ask them what a triomino looks like. We decide that there are only two. The other triominos are congruent to these two, by rotation, reflection and translation. I then ask my students to find all possible tetrominos. I suggest you try it before reading on.

When I first gave this activity, many years ago, I learned that it took about ten minutes with several interesting and useful discussions about double-counting and that there would be several students who did not discover all five. Eventually, those students did find all five and were ready for the problem I really wanted them to tackle: finding all pentominos. I then asked them if they could construct a square out of the pentominos, and if not, why not? What rectangles could they construct? It is a nice activity. But the nice activity is not the point of this blog.

One year my students immediately solved the tetrominos problem, and I mean immediately. Like in fifteen seconds or as long as it took them to write the answer. This task was not a difficult problem for them at all. This class was not particularly exceptional, and I was bewildered, until I heard one of the students ask another how he did it so fast. His answer: Tetris.

What had happened from one year to the next was the game whose pieces were these pieces. What was once difficult, was now simple.

I have often thought about what makes a problem difficult and another problem simple. I am particularly interested in problems that some students seem to understand with little effort and other students struggle with. I think of this class. I am convinced that what we perceive as talent and insight is often just experience and familiarity with similar ideas.

I ask my honors students how many of them played with Lego when they were little. Many of them giggle and smile. Some admit that they still play with Lego. Often the conversation moves to other toys of the same nature. When I ask the same questions of students who struggle with math, I get blank stares. I think we need to pay careful attention to what sort of experiences our students have had. Even more important, I think we need to build in lessons so that they are previewing topics we know will be difficult later. And they can't just be pencil and paper activities.

Peg Cagle, a middle school teacher from California, gave a fascinating talk about how students are having different experiences growing up than we had. She asked her students, I think she said about 180, how many had climbed a tree. None had. I think that effects their sense of what three dimensional space looks like and feels like.

Sunday, May 8, 2011

A theme of this blog for several months has been the importance of challenging problems to help students learn. Last week P.J. wrote about one of the most challenging problems every teacher faces: those students who just don't get it, and whom we can't get to no matter what we try. Now that I am retired and am not faced with that particular problem on a day to day basis, I may be able to provide a different perspective.Of course I agree with PJ: the most important thing is to NEVER give up on the student, no matter how hopeless it seems. Learning is complicated and unpredictable. Some time ago, I was tutoring a student on a weekly basis, one-on-one, for an hour at a time. She was taking seventh-grade math, and it seemed hopeless. I would give her a problem. She would get it wrong. We would go over it. I would give her another. She'd get it. We'd go on to the next topic. Fifteen minutes later, we'd return to the first topic, and she would get it wrong. This pattern happened every session, every week. In eighth grade, a similar experience presented itself as she studied algebra. When high school arrived, she took algebra again, this time at an honors level. She not only passed, but was disappointed when she only got an A-. She managed a B in Geometry Honors and ended up taking four years of math, the last two at a regular level, and going to a good college.She had resources, and she had encouraging parents who did not give up on her, and she was determined to succeed--but she did succeed, and that is the thing to remember. Each person has his or her own learning personality. Each person can learn, but some in radically different ways, and many are looking for an excuse to quit trying. A teacher who will not give up on the student fights that student's urge to quit.In her outstanding book, Overcoming Math Anxiety, Shelia Tobias points out that virtually every adult she interviewed had a vivid memory of a moment when a teacher made the interviewee believe that math was not possible for that person to learn. We must ensure that every student believes that we believe that student can learn important mathematics, somehow, sometime. One of my colleagues, Janet Webb, used to continually remind us that every parent sends us the best child they have and expects us to do our best to educate that child.Another related thought is that the specific content of the course is less important then the intangible things that the student takes away. The impact of their time in your class has more to do with the way you treated them and the respect, enjoyment and excitement about mathematics you demonstrated than the actual algorithms and theorems they were tested on.

Monday, May 2, 2011

This is a pessimistic entry, but it encapsulates some issues I've been gnawing on for a while.
Every year, I see a few students who just can't seem to pull it together, and eventually sink faster and faster. But when I try to help them, I find that more often than not, the problems are deeper than I first thought. These students aren't just unprepared, or unwilling to work. Most often, the problem is a combination:

On a purely mathematical level, their skills are weak, and their conceptual understanding is even weaker. They pull out a calculator to subtract 72 from 180, can't measure an obtuse angle with a protractor, and believe that when the sides of an angle are extended, the angle's measure increases.

More generally, they have trouble understanding complex tasks. For example, one student, when asked to draw four quadrilaterals, measure their angles, and compute their angle sum, instead freehanded four rectangles. The student didn't realize that shapes without straight lines have angles that are at best poorly-defined; didn't realize that the experiment is essentially pointless with rectangles anyway; and, when he got an angle sum greater than 360 degrees, didn't try to resolve the discrepancy.

On the level of practical reasoning, they have trouble connecting present behavior to future results, especially when positive results require sustained effort. Because of their poor preparation, on the occasions when they do put in effort, they don't get good grades on tests or quizzes. They quickly learn that these sporadic efforts don't get them the results they want, and then decide that those efforts were not worthwhile. A student gets a 15% on a quiz; after lots of studying, the test score is a 60%, and instead of seeing a 300% gain, the student says "I studied and I still got a D, so why bother?" Who would blame him?

They don't have the support of families who can help them with the mathematics, or even help them connect their efforts (or lack of efforts) to results in any concrete way. These parents may not come to parent-teacher conferences, or if they do come, profess an inability to actually change what their children are doing. I tend to believe them: as a parent myself, I've come to realize how hard it is to actually make anyone do anything.

While I've spent hours thinking about what causes these different deficiencies, fundamentally that thinking doesn't help those students. But I haven't had much more success figuring out what to do about them. They need lots and lots of scaffolding: in math and in academic skills generally. They need our help connecting the dots from incremental efforts to incremental gains, until their gains become large enough to be visible to the naked eye. And they need to trust the very teacher who--in their eyes--is asking the unreasonable and punishing them for failing to accomplish the apparently-impossible. Finally, they need all those things on a sustained basis, for weeks and months, rather than days. And then we wonder why they fail.

Personally, I'm able to help one or two of these kids per year, to the point where they're actually reasonably successful: passing classes, not eternally frustrated. But I have at least half a dozen, and that's at a super-selective public high school. I don't see any way to increase my own capacity, both because success requires so much time and energy per student, and because which students I "connect" with seems, at this point, a matter of luck more than anything else.

Fellow teachers: have I missed anything? Any thoughts on a way to do this better?

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About Us

John Benson taught for forty-two years as a classroom teacher in public schools. The first year was at Frederick Douglass High School in Atlanta, Georgia, and the next forty-one years, at Evanston Township High School in Evanston, Illinois. He has been actively involved in many math organizations and attributes much of his success as a teacher to this participation. He was founder and president of the North Suburban Math league, spent twelve years grading AP Calculus exams, and taught for many summers at the Center for Talent Development at Northwestern University. He was the Presidential Awardee in Mathematics for Illinois in 1987. He has been a contributing author for four secondary math textbooks, most recently the third edition of UCSMP Geometry.

P.J. Karafiol has taught math for 18 years: at Phillips Academy in Andover, Massachusetts; at Providence-St. Mel School on Chicago's West Side; and for the last twelve years, at Walter Payton College Prep High School in Chicago, where he is the Coordinator for Curriculum, Instruction, and Assessment. Like John, he attributes any success he has had to reflection on past failures and to discussions with his friends and colleagues. He is the lead author of the ARML competition and a co-author of many other contests. He was the Presidential Awardee in mathematics in Illinois in 2009. He is the co-author of two textbooks, the third editions of UCSMP Advanced Algebra and Functions, Statistics, and Trigonometry.